Algebra · Chapter 13

Speed, Distance, Time

Race, chase and meet: use distance = speed × time, read distance–time graphs, and dodge the average-speed trap.

What's this chapter about?

13.1 Speed, Distance, and Time
13.2 Practice Problems

You'll need

13.10%

Speed, Distance, and Time

In this section you will be able to use distance = speed × time to find any one of the three, and find average speeds and meeting times.

Sohil and Sejal live 12 miles apart. They start running toward each other at the same moment. Sohil runs 2 miles per hour and Sejal runs 4. Where will they meet?

Every problem in this chapter uses one rule: distance = speed × time. If you know two of the three, you can find the third.

The graph under the track shows position against time. A steeper line means a faster runner. When two lines cross, the runners are at the same place at the same time: they meet!

Look:watch your units. A speed in miles per hour gives a time in hours. Multiply by 60 to get minutes.

Exploreplay first, formulas later

Try this: Sohil and Sejal start 12 miles apart and run toward each other. Before you press Play, predict: when and where do they meet?
Try this: Press Play and watch the graph. Where do the two lines cross? Read off the time and the place.
Try this: Make Sejal faster. Does the meeting point move toward Sohil or away from him? Why?
Try this: Turn on Show the formula. Why do you add the two speeds when they run toward each other?

Name it

Theorem 13.1.1 · Distance, Speed, and Time

\text{Distance} = \text{Speed} \times \text{Time}

Equivalently,

\text{Speed} = \frac{\text{Distance}}{\text{Time}} \qquad \text{Time} = \frac{\text{Distance}}{\text{Speed}}

Theorem 13.1.2 · Average Speed

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Remark 13.1.4

Often, we may have to form equations with distance, speed, and time and use them to solve our problem. Give the unknown a letter, write $\text{distance} = \text{speed} \times \text{time}$ for each traveller, and connect the equations with what the problem tells you (same time, same distance, a gap that closes…).

Watch itpredict, then reveal

Example 13.1

Joe needs to get to his office 40 miles away. If he drives at 60 miles per hour, how many minutes will it take him to get to work?

1
What information do we know?

Example 13.2

Omega Learn

Sohil is running along a 2 mile path at 8 miles per hour and back on the same path. On the way back he is tired so for after every 1 mile he runs on the way back, his speed is instantly reduced by 1 mile per hour. What is his average speed throughout the whole trip?

1
How long does the trip out take, in hours?

Predict first:
Your prediction

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Example 13.3

AMC 8

Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?

1
Linda travels for $60$ minutes. If one mile takes her $m$ minutes, how many miles does she go?

Predict first:

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Example 13.4

AMC 8

A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?

1
Call Zia's starting stop $0$ , so the bus starts at stop $-3$ . Each stop costs the bus $2 + 1 = 3$ minutes. At what minute does the bus reach stop $1$ ?

Predict first:
Your prediction

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Example 13.5

AIME

Points $A$ , $B$ , and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$ , Paul starting at $B$ and running toward $C$ , and Eve starting at $C$ and running toward $A$ . When Paul meets Eve, he turns around and runs toward $A$ . Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$ .

1
Let Eve's speed be $v$ . Paul's speed is how many times Eve's?

Predict first:
Your prediction

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Try it

Problem 13.2.4

Omega Learn Math Contest

Jeff is driving 250 miles to Orlando at a constant speed of 50 miles per hour. He needs to reach in exactly 5 hours. After driving 100 miles, he takes a 30 minute break. He then continues driving at a new constant speed that would allow him to arrive at the same time. How much faster did he drive after his break compared to before the break in miles per hour?

Problem 13.2.8(additional)

Joe drives at a speed of 30 miles per hour for 60 miles. He also drives at 45 miles per hour for 15 miles. Find the average speed of his car.

Problem 13.2.9(additional)

MATHCOUNTS

Jones is chasing a car 800 meters ahead of him. He is on a horse moving at 50 km/h. If Jones catches up to the car in 4 minutes, how fast was the car moving in km/h?

Contest cornerreal contest problems

Problem 13.2.1

AMC 8

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

Problem 13.2.3

AMC 8

Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers $2\frac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella?

Problem 13.2.5

AMC 8

Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?

Problem 13.2.6

AMC 8

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?

Problem 13.2.10(additional)

AMC 8

After school, Maya and Naomi headed to the beach, 6 miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

Checkpoint for section 13.1

Get 2 of 3 right to master this section. One try each, no hints.

13.2

Practice Problems

In this section you will be able to dodge the average-speed trap and set up equations for escalators, ships and long chases.

You drive to the beach at 60 km/h and drive home at 30 km/h. Was your average speed 45 km/h? Almost everyone says yes. Almost everyone is wrong!

Average speed is total distance ÷ total time. You spend more time at the slow speed, so the slow speed counts more.

These problems are tougher. Give each traveller a letter for the unknown, write distance = speed × time for each one, and connect the equations. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Exploreplay first, formulas later

Try this: You drive 60 km out at 60 km/h and 60 km back at 30 km/h. Guess the average speed, then press Check my guess. Did you say 45?
Try this: Press Drive! and watch the time bar. Which part of the trip lasts longer?
Try this: Set the Back speed to 60 km/h too. Now does the mean of the speeds give the right average? Why?

Contest cornerreal contest problems

Problem 13.2.2

AMC 10

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at 4 miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to 2 miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at 3 miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

Problem 13.2.7

AIME

Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)

Problem 13.2.11(additional)

AMC 10

David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?

Problem 13.2.12(additional)

EMCC

Jacob and Alexander are walking up an escalator in the airport. Jacob walks twice as fast as Alexander, who takes 18 steps to arrive at the top. Jacob, however, takes 27 steps to arrive at the top. How many of the upward moving escalator steps are visible at any point in time?

Problem 13.2.13(additional)

AMC 10

Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts 210 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts 42 steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?

Problem 13.2.14(additional)

AIME

Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at 6, 4, and 2.5 miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are $n$ miles from Dodge, and they have been traveling for $t$ minutes. Find $n + t$ .